3.344 \(\int \frac{x^3}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 a^3}{b^4 \sqrt{a+b x}}+\frac{6 a^2 \sqrt{a+b x}}{b^4}-\frac{2 a (a+b x)^{3/2}}{b^4}+\frac{2 (a+b x)^{5/2}}{5 b^4} \]

[Out]

(2*a^3)/(b^4*Sqrt[a + b*x]) + (6*a^2*Sqrt[a + b*x])/b^4 - (2*a*(a + b*x)^(3/2))/
b^4 + (2*(a + b*x)^(5/2))/(5*b^4)

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Rubi [A]  time = 0.0495814, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 a^3}{b^4 \sqrt{a+b x}}+\frac{6 a^2 \sqrt{a+b x}}{b^4}-\frac{2 a (a+b x)^{3/2}}{b^4}+\frac{2 (a+b x)^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x)^(3/2),x]

[Out]

(2*a^3)/(b^4*Sqrt[a + b*x]) + (6*a^2*Sqrt[a + b*x])/b^4 - (2*a*(a + b*x)^(3/2))/
b^4 + (2*(a + b*x)^(5/2))/(5*b^4)

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Rubi in Sympy [A]  time = 10.9244, size = 63, normalized size = 0.95 \[ \frac{2 a^{3}}{b^{4} \sqrt{a + b x}} + \frac{6 a^{2} \sqrt{a + b x}}{b^{4}} - \frac{2 a \left (a + b x\right )^{\frac{3}{2}}}{b^{4}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**(3/2),x)

[Out]

2*a**3/(b**4*sqrt(a + b*x)) + 6*a**2*sqrt(a + b*x)/b**4 - 2*a*(a + b*x)**(3/2)/b
**4 + 2*(a + b*x)**(5/2)/(5*b**4)

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Mathematica [A]  time = 0.0239952, size = 45, normalized size = 0.68 \[ \frac{2 \left (16 a^3+8 a^2 b x-2 a b^2 x^2+b^3 x^3\right )}{5 b^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x)^(3/2),x]

[Out]

(2*(16*a^3 + 8*a^2*b*x - 2*a*b^2*x^2 + b^3*x^3))/(5*b^4*Sqrt[a + b*x])

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Maple [A]  time = 0.009, size = 42, normalized size = 0.6 \[{\frac{2\,{b}^{3}{x}^{3}-4\,a{b}^{2}{x}^{2}+16\,{a}^{2}bx+32\,{a}^{3}}{5\,{b}^{4}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^(3/2),x)

[Out]

2/5/(b*x+a)^(1/2)*(b^3*x^3-2*a*b^2*x^2+8*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.34211, size = 76, normalized size = 1.15 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b^{4}} - \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} a}{b^{4}} + \frac{6 \, \sqrt{b x + a} a^{2}}{b^{4}} + \frac{2 \, a^{3}}{\sqrt{b x + a} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

2/5*(b*x + a)^(5/2)/b^4 - 2*(b*x + a)^(3/2)*a/b^4 + 6*sqrt(b*x + a)*a^2/b^4 + 2*
a^3/(sqrt(b*x + a)*b^4)

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Fricas [A]  time = 0.209178, size = 55, normalized size = 0.83 \[ \frac{2 \,{\left (b^{3} x^{3} - 2 \, a b^{2} x^{2} + 8 \, a^{2} b x + 16 \, a^{3}\right )}}{5 \, \sqrt{b x + a} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

2/5*(b^3*x^3 - 2*a*b^2*x^2 + 8*a^2*b*x + 16*a^3)/(sqrt(b*x + a)*b^4)

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Sympy [A]  time = 8.93163, size = 1538, normalized size = 23.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**(3/2),x)

[Out]

32*a**(45/2)*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**
2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**1
0*x**6) - 32*a**(45/2)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 10
0*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6
) + 176*a**(43/2)*b*x*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18
*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a
**14*b**10*x**6) - 192*a**(43/2)*b*x/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*
b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a*
*14*b**10*x**6) + 396*a**(41/2)*b**2*x**2*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**
19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a
**15*b**9*x**5 + 5*a**14*b**10*x**6) - 480*a**(41/2)*b**2*x**2/(5*a**20*b**4 + 3
0*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 +
30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 462*a**(39/2)*b**3*x**3*sqrt(1 + b*x
/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 +
75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 640*a**(39/2)*b
**3*x**3/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x
**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 290*a**(37
/2)*b**4*x**4*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x*
*2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**
10*x**6) - 480*a**(37/2)*b**4*x**4/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b*
*6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**1
4*b**10*x**6) + 92*a**(35/2)*b**5*x**5*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*
b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**1
5*b**9*x**5 + 5*a**14*b**10*x**6) - 192*a**(35/2)*b**5*x**5/(5*a**20*b**4 + 30*a
**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30
*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 16*a**(33/2)*b**6*x**6*sqrt(1 + b*x/a)/
(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*
a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 32*a**(33/2)*b**6*x
**6/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 +
75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 6*a**(31/2)*b**
7*x**7*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 10
0*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6
) + 2*a**(29/2)*b**8*x**8*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a
**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 +
5*a**14*b**10*x**6)

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GIAC/XCAS [A]  time = 0.204771, size = 82, normalized size = 1.24 \[ \frac{2 \, a^{3}}{\sqrt{b x + a} b^{4}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{5}{2}} b^{16} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{16} + 15 \, \sqrt{b x + a} a^{2} b^{16}\right )}}{5 \, b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

2*a^3/(sqrt(b*x + a)*b^4) + 2/5*((b*x + a)^(5/2)*b^16 - 5*(b*x + a)^(3/2)*a*b^16
+ 15*sqrt(b*x + a)*a^2*b^16)/b^20